Despite dramatic advances in the energy resolution and throughput of electron monochromators for high resolution electron energy loss spectroscopy (HREELS), a major limitation of conventional, dispersive sector, electron energy analyzers is that they are inherently serial devices, leading to long data acquisition times. The advantage of higher resolution leads to trade-offs in performance (throughput) because channel step size must be reduced, and therefore increasing the number of channels required to measure a given spectral region. Using a multi-channel plate detector to ameliorate this problem is one possibility. Indeed, time-resolved HREELS measurements have been demonstrated with a multi-channel plate in the dispersive plane of a conventional analyzer. However, parallel detection can be accomplished in this way only over a limited spectral range without degrading resolution. Thus, development of an analyzer based upon parallel detection would benefit both typical spectral investigations and allow new experiments to be performed, such as recent inelastic diffraction experiments which are both momentum and energy resolved.
Pseudo-random binary sequences (PRBS), also known as maximal length (ML) shift register sequences and/or pseudorandom noise (PN) sequences, have been used for modulation of photon and particle beams in widely used time-of-flight (TOF) techniques. These find application in, for example, neutron scattering, molecular beam scattering, and ion mass spectroscopy. The PRBS-TOF method achieves a throughput advantage over single pulse TOF due to the 50% duty cycle. PRBS modulation has been combined with TOF-MS, for example in a paper by Brown, W. L., et al., xe2x80x9cElectronic sputtering of low temperature molecular solids,xe2x80x9d in Nuclear Instruments and Methods in Physics Research, Vol. B1, 1984, pp. 307-314. In this example, an incident ion beam was pulsed with a pseudo-random sequence. The ion beam impinged upon a condensed water matrix sample, sputtering or producing secondary ions and neutrals which are measured in a time-of-flight detector. An electron impact ionizer was used to ionize the neutral and then a quadrupole filter was employed to mass select the products. Thus, in this case, the TOF technique was used to measure the energy distribution. To improve the signal-to-background ratio, PRBS modulation and cross-correlation recovery techniques were used, assuming that the source modulation was ideal.
More specifically, in such approaches, the underlying TOF spectrum (the object spectrum, o) is modulated with the PRBS sequence, p, resulting in a periodic, time sequence that is assumed to be defined mathematically as, (p{circle around (xc3x97)}o). In the standard cross-correlation recovery method, an estimate of the TOF spectrum, r, is obtained by correlating the detected TOF signal data with the PRBS modulation sequence, p: r=p⊕(p{circle around (xc3x97)}o). Here, {circle around (xc3x97)} and (⊕ denote convolution and correlation, respectively.
A special property of maximal length PRBS sequences is that the autocorrelation of the discrete binary sequence p is a substantially a delta function; therefore, the recovered spectrum, r, is substantially identical to the original object spectrum, o. In reality, the modulation function p is continuous, but r is an estimate of o as long as the time base (minimum pulse width) of the modulation function is small compared to the linewidth of the narrowest features in o. If this is not the case, then the throughput advantage is gained at the expense of resolution in the recovered spectrum, and over-sampling of the modulated signal, (p{circle around (xc3x97)}o), leads to a recovered spectrum which is the autocorrelation (p⊕p) (roughly, a triangular pulse) convoluted with the object function: r=(p⊕p){circle around (xc3x97)}o.
In fact, the modulation of the particle beam, whether performed at the source, with a spinning disk type of mechanical chopper, or with an electrostatic deflection based device, is at best described approximately as a convolution with the ideal sequence, (p{circle around (xc3x97)}o). First, the actual effect of the modulating device on the particle beam differs to some extent from the ideal sequence, p. A number of artifacts in the recovered object function, r, are well known in the art, and some types of non-ideal behavior can be corrected through post processing when (p⊕p) differs from a delta function, such as arises from machining errors in creating the slots in mechanic spinning disks. Second, most modulators do not act in exactly the same manner on different particles in the beam; for example, the finite thickness of spinning disks leads to a velocity dependent modulation function in molecular beam scattering applications. In this case, the assumption of a convolution is not strictly true.
To the extent that the modulation can be described by a convolution, and the actual modulation function, p, is known or can be estimated, the object function may be recovered simply by Fourier deconvolution. In practice, the presence of noise in measured data complicates deconvolution of spectral data in the simplest cases when the instrument function can be described by a single feature.
The deconvolution of a PRBS modulation sequence, in which the data contains multiple overlapping copies of the underlying object function, has not been reported in spectroscopic applications, to our knowledge.
Probability-based estimation methods for recovery of one-dimensional distributions, and for resolution enhancement of one-dimensional spectral data and two-dimensional image data, have been used by astronomers since 1972. (See Richardson, W. H. 1972, xe2x80x9cBayesian-Based Iterative Method of Image Restorationxe2x80x9d, J. Opt. Soc. Am. 62, 55-59; Frieden, B. R. 1972, xe2x80x9cRestoring with Maximum Entropy and Maximum Likelihoodxe2x80x9d, J. Opt. Soc. Am. 62, 511-18; Lucy, L. B. 1974, xe2x80x9cAn iterative technique for the rectification of observed distributionsxe2x80x9d, Astron. J. 19, 745-754; and Ables, J. G. 1974, xe2x80x9cMaximum Entropy Spectral Analysisxe2x80x9d, Astron. Ap. Suppl. 15, 383-93.) Recent success with iterative maximum likelihood and Bayesian methods has been demonstrated in a paper by Frederick, B. G., et al., entitled xe2x80x9cSpectral restoration in HREELS,xe2x80x9d in the Journal of Electron Spectroscopy and Related Phenomena, Vol. 64/65, 1993, pp. 825. The maximum likelihood result is simply an array, which convoluted with the modulation function, fits the data as well as possible, given the noise distribution. A well known example of this approach is the algorithm reference in the paper by L. B. Lucy. The Bayesian method employed by Frederick, et al., includes a maximum entropy constraint that limits the degree of resolution enhancement in a manner that leads to a single converged estimate with no arbitrary adjustable parameters.
We have invented an instrument that uses a PRBS particle beam modulator and detector together with a probability based estimation algorithm for removing artifacts introduced by components of the instrument.
Specifically, in a preferred embodiment, an interleaved comb-type chopper can modulate an electron beam with rise and fall times of less than a nanosecond, which corresponds to meV energy resolution for low energy electrons. The finite penetration of the fields associated with this electrostatic device produces certain non-ideal behavior, which we characterize in terms of an xe2x80x9cenergy corruptionxe2x80x9d effect and a lead or lag in the time at which the beam responds to the chopper potential.
According to our invention, for the first time, an instrument employs maximum likelihood, maximum entropy, or other probability based estimation methods to recover the underlying TOF spectrum, in spite of the corruption. These methods can be used to undo the corrupting effects of (a) less than perfectly xe2x80x9cmaximal lengthxe2x80x9d PRBS sequence; (b) specific chopper effects; and (c) in general, detected signal artifacts introduced by components of the instrument.
Compared to the standard cross correlation method, i) the resolution is improved relative to the nominal time base resolution of the PRBS or other modulating sequence; ii) the Poisson (pulse counting) noise is accounted for; and iii) artifacts associated with imperfections of the chopper or other component performance are reduced.
A spectroscopy instrument thus makes use of statistical estimation techniques to account for component artifacts in accordance with the present invention. The instrument may use several different types of physical phenomena to determine the attributes of a sample. In one specific embodiment, a particle source such as ion source provides a stream of particles to a propagation path. The instrument uses a modulator grid or xe2x80x9cchopperxe2x80x9d, driver electronics, and a sequence generator to modulate the ion source. The ion source may be modulated directly either prior to or subsequent to its application to a sample in order to provide a particle beam that is modulated in time.
The modulator may itself take different forms; one particularly useful implementation as a grid of wires. In addition, spinning disk-type modulators can be utilized which encode the specific modulation sequence as a series of holes around the periphery.
Particles of different chemical makeup exhibiting different physical time-of-flight properties thus travel down the propagation path at different times over different distances to arrive at one or more detectors. A time-to-digital converter then provides a signal to a computer to analyze the detected signal to determine the chemical makeup of the sample.
As has been alluded to above, the computer uses a component model that makes use of maximum likelihood estimation. In an implementation of a statistical method that uses a maximum likelihood method, the so-called Lucy algorithm can be used to refine the estimate for the object spectrum. It will be understood by those of skill in the art that other algorithms can be used.
The computer may perform this statistical method as follows. For example, a system response function is first chosen. The system response function may be an a priori measured response, such as the modulated signal measured with a monochromatic source or from a monochromator. It may also be obtained from a theoretical model, or from a set of data measured on the same sample, for example a high resolution single pulse TOF spectrum and a PRBS modulated spectrum. If the single pulse spectrum is a good estimate of the underlying object spectrum, o, then the PRBS modulated data, y, and then an estimation method such as Lucy can be used to obtain p by deconvolution of y with the estimated object spectrum, o.
The computer also obtains an initial estimate, oi, of the object spectrum. This can be from a previous spectrum or from performing a cross-correlation of the modulating sequence with system response data.
The system response and initial object spectrum estimate are then combined with a model of the instrument that may for example include the noise and physical characteristics of the instrument, to select an appropriate probability based estimation algorithm. A refined estimate is then obtained; the estimate obtained may be acceptable as determined by criteria, or iteration via the refinement process may be necessary.
Thus, although PRBS modulation has been known and used in many areas of spectroscopy in the prior art {including neutron scattering, molecular beam scattering, TOF-MS and secondary ion mass spectroscopy (SIMS)}, and although digital signal processing methods for data recovery have been utilized in an even wider range of spectroscopies, we know of no examples in which the actual response function of the system, particularly the modulator, has been estimated and used to directly deconvolute the PRBS modulated data. There are a number of reasons that deconvolution of PRBS modulated data may not have been contemplated in this field. This may have been driven by the fact that the delta-function autocorrelation properties of PRBS sequences were presumed to provide perfect recovery of the underlying spectrum, and that no further artifacts were introduced by the process.
There has in fact been a general skepticism towards deconvolution of one dimensional spectroscopic data, in part due to the difficulty of the inversion problem. The presence of noise in real data leads to artifacts, even when the response function is known accurately. Filtering usually involves adjustment of some arbitrary parameters, such that the estimate obtained is not unique and the results are subjective. Many methods require that assumptions be made about the underlying spectrum, such as the shape and number of peaks. Furthermore, in the case of PRBS modulation, in which the data contains multiple, overlapping copies of the desired object spectrum, it is not obvious, a priori, that
i. there is sufficient phase information to allow deconvolution of the data, even for the case in which the response function is known and the data is measured without noise; or
ii. that existing algorithms may not converge to the true solution.
There has been an emphasis upon real-time display during data acquisition, such that Fourier transform based instruments did not become popular until Fast Fourier transform algorithms and sufficiently fast computers became available. The iterative methods we have utilized here have required sufficient computational power that real-time processing during data acquisition has been a limitation; nevertheless, dramatic increases in computational power associated with DSP""s and FPGA""s now allow much more sophisticated processing in real-time.
Unlike traditional non-linear least squares fitting algorithms that optimize typically not more than 15 or 20 parameters, the methods we have chosen require optimization of at least as many parameters as there are points in the modulated time series data. The method makes no assumptions regarding the number or shape of features in the underlying TOF spectrum, except that the spectrum is positive definite. Therefore, the inversion problem appears to be much more difficult than tradition non-linear fitting problems.
A critical factor in our approach is to oversample the data, relative to the PRBS time unit, which is counter to the prevailing practice in the field. Brock et al, in U.S. Pat. No. 6,300,626 note that xe2x80x9cThis procedure will increase the definition of individual peaks, but is not able to increase the time or mass resolution of the device.xe2x80x9d While this is true for the measured time resolution, particularly for a single pulse TOF spectrum, the information content in the signal may be significantly enhanced by oversampling the data and the system response function. In addition to dramatically reducing artifacts in the recovered spectrum due to certain kinds of non-ideal behavior of the modulator, we have demonstrated that a resolution enhancement by a factor of at least 8xc3x97 can be achieved with PRBS modulation. This is in part due to the square pulse like shape of the response function, retaining relatively high frequency components in the Fourier domain.